Introduction

Multicellular pattern formation is a recurring theme in the life sciences, exemplified by traveling waves of neural activity1 and cardiac contraction2, or standing waves of coat-pigment patterns3. These patterns are typically realized through intercellular communication via activation and/or inhibition signals4. The heterogeneity of cellular conditions, including cell types, reactivities, and densities, is a critical determinant in controlling their dynamics5. Theoretical studies, particularly those focusing on systems generating Turing patterns, have examined heterogeneity at both micro- and macro-scales, such as the salt-and-pepper randomization of cells for their initial conditions3 and morphogen gradients spanning the entire system6. These heterogeneities enhance the complexity of system dynamics by producing labyrinth patterns and converting them into stripe patterns, which are not observed in homogeneous systems. Despite these insights, mesoscale heterogeneity between the scale of cells and the entire system has received limited attention. The effects of mesoscale heterogeneity on wave pattern formation have been examined in few simulation studies on typical signal-relaying systems for simple traveling waves (excitable systems)7,8. These studies considered the spatially different signal-relaying ability (excitability) at a size similar to traveling waves as mesoscale heterogeneity. Consequently, the systems generated spiral waves that did not develop in homogeneous systems spontaneously, demonstrating the potential importance of mesoscale heterogeneity in wave pattern formation. However, the significance of mesoscale heterogeneity in real biological systems remains underexplored, likely owing to the lack of suitable experimental models and technical challenges in the functional analysis of mesoscale heterogeneity, where the simultaneous detection of mesoscale heterogeneity and system dynamics in living samples is essential.

The social amoeba, Dictyostelium discoideum, provides an excellent model for studying the functional importance of mesoscale heterogeneity in wave pattern formation, because we believe that these two issues can be resolved in this system. Upon nutrient starvation, the whole-system consisting of 104,5,6 cells develops spiral-shaped sub- to a few millimeter aggregation waves9. This process is driven by the intercellular relay of the chemoattractant cyclic adenosine monophosphate (cAMP)10,11 whose reaction-diffusion dynamics can be explained by a simple combination of local synthesis, degradation, and diffusion12,13,14. Beyond these general properties of signal-relaying systems, D. discoideum cells possess two notable features that make them suitable for investigating mesoscale heterogeneity: one feature suggests the presence of mesoscale heterogeneity, and the other contributes to overcoming technical challenges in their analysis. The first feature is the developmental transition of wave propagation from a silent and non-signaling state at early development to spatially homogeneous propagation at late development. During this transition, we previously observed spatially isolated pulsing cells surrounded by non-pulsing cells, which resulted in a collapsing wave at the sub-millimeter scale15. This suggests the presence of mesoscale heterogeneity in cellular excitability, which provides a structural scaffold for collapsing waves. The second feature is the positive feedback development of cellular ability to relay cAMP signals. Since gene expression controlling cAMP dynamics is positively regulated by cAMP signaling itself11,16,17, cells experiencing more rounds of cAMP signaling should acquire a higher ability to relay these signals. This indicates that the cellular excitability can be evaluated based on the number of times they experience cAMP signals. Based on this idea, we considered the spatial distribution of excitable cells to represent the local excitability that acts as a structural scaffold for cAMP waves. Such an indirect but straightforward evaluation of local excitability aligns well with cAMP wave analysis by live imaging, allowing simultaneous investigation of how wave patterns are influenced by structural scaffolds, if they exist.

In this study, we aimed to understand the role of mesoscale heterogeneity in pattern formation using developing D. discoideum. We demonstrate spatially heterogeneous wave propagation on the mesoscale heterogeneity in excitability, characterized by a newly introduced pulse-count analysis. Specifically, we identified the wave rotation around the low-excitability core occurring on mesoscale heterogeneity, which is key to spiral pattern formation. Together with the perturbation experiments on mesoscale heterogeneity, we conclude that mesoscale heterogeneity governs pattern formation in developing D. discoideum.

Results

Spatially heterogeneous cAMP wave propagation

To detect cellular cAMP ([cAMP]in) in developing D. discoideum cells, we established a strain expressing a fluorescent indicator for cAMP (Flamindo218,19 fused with mRFPmars20). Conventional microscopy is inadequate for imaging the entire process of cAMP wave development because its field of view (FOV), a few millimeters in size, is substantially small to capture whole spiral waves with single-cell resolution. Instead, we utilized a recently innovated trans-scale imaging system named AMATERAS15, whose exceptionally large FOV at the centimeter scale containing 1.3 × 105 cells is essential to capture the emergence of multiple spiral waves (Fig. 1a). The spatial (2.3 μm) and temporal (30 s) resolutions were high enough to conduct single-cell tracking for the quantitative analysis of cAMP wave with dissecting the timing of cAMP pulsing among closely positioned cells (Fig. 1b–c).

Fig. 1
figure 1

Spatially heterogeneous propagation of cAMP wave. (a) One-shot imaging of approximately 1.3 × 105 cells at single-cell resolution. Composite image of Flamindo2 (Green) and mRFPmars (Red). (b) Close-up view of a region indicated by a white box in (a) at 7:00. Tracked single cells are indicated by open circles whose colors represent peak timings of cAMP pulses. Mesh size for analysis regions of interest (ROIs), 110 μm. (c) Time course of cAMP pulses for representative cells in (b). (d,e) Kymographic representation of cAMP waves (d) and its spatial occupancy (e) at the white broken line in (a). (f) Change in the spatial occupancy of cAMP pulsing with time.

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To determine whether cAMP wave propagation was spatially heterogeneous or homogeneous, we first analyzed the kymograph of cAMP waves from their onset to the terminal stage of spiral development. For simplicity, the images were spatially coarse-grained into 110 × 110 μm2 regions of interest (ROIs) containing approximately 10 cells, and ROI-level pulsing was identified if more than two cells exhibited fluorescent signal changes for cAMP pulsing. Local pulsing (Fig. 1d white) and its sum over time (Fig. 1e) were binarized to illustrate the pulsing pattern and field coverage, respectively. The kymograph showed that localized cAMP pulsing began in a few ROIs 5 h after starvation (5:00), followed by a gradual increase in the pulsing ROIs until all ROIs became pulsatile after 7:30. Spiral waves seemed to appear around 7:00, since the spatiotemporal gap of cAMP pulsing, which is a signature of spiral waves in the kymograph21 (Supplementary Fig. 1a–b), was evident (Fig. 1d yellow box). The time evolution of the spatial occupancy of cAMP pulsing showed that the pulsing ROIs increased from 10 to 90% between 5:00 and 7:00 (Fig. 1f). The binary representation of the kymograph for the pulsing ROIs summed over time revealed the development of clusters of pulsing ROI in a non-pulsing background (Fig. 1e). At 6:30, 15 localities consisting of 2–20 pulsing ROIs were identified, whose sizes correspond to 0.2–2 mm on the analysis line in the kymographs (Fig. 1e green), demonstrating that spatially heterogeneous wave propagation occurred before spiral wave emergence. These observations also suggest the presence of a yet-unidentified spatially heterogeneous wave permissivity, whose length scale is expected to be sub to a few millimeters.

Mesoscale heterogeneity in the local excitability visualized by pulse-count analysis

Spatially heterogeneous wave propagation during early and middle development (Fig. 1d–e) can occur at spatially different cell densities. However, as the cellular distribution was nearly homogeneous during these stages in our observation (4:00–8:00, Supplementary Fig. 1c–d), other contributing factors must be investigated. Therefore, we hypothesized the presence of structural scaffolds with different excitabilities for spatially heterogeneous wave propagation. This can be visualized if the local excitability is measurable; however, in practice, there are no suitable molecular markers to measure it. Instead, we focused on the cellular history of how many times the cells experienced cAMP pulsing (pulse-count) as a potential marker for local excitability because it has been well accepted that the gene expression controlling cAMP signaling is positively regulated by cAMP signals itself16,17. Indeed, the cAMP pulsing amplitudes and inter-pulse intervals, which are phenotypic indicators of gain and recovery from refractoriness, respectively, were higher and shorter for cell population experiencing more cAMP pulsing rounds (Supplementary Fig. 2). Thus, we decided to use the pulse-count to study local excitability.

To demonstrate the utility of the pulse-count in representing local excitability, we focused on a representative area of 440 × 440 μm2 containing approximately 200 cells that repeatedly emit cAMP pulses (Fig. 2a), and detected pulsing cells with their pulse-counts. By the end of the fourth wave propagation in this area, 14 of the approximately 200 cells had experienced 1–4 cAMP pulsing rounds (Fig. 2b). The temporal sequence of cAMP pulsing in the fourth wave began in the cell with the highest pulse-count, followed by those with two and one pulse-counts, while the others remained at zero pulse-count (Fig. 2b), illustrating a locally collapsing wave. Such a collapsing wave pattern was also observed at the end of the fifth and tenth wave propagations, when the number of pulsed cells increased to 35 and 82, respectively (Fig. 2c–d). Newly pulsed cells were found at both the outer border and inside the cluster, which contributed to strengthening cAMP wave propagation. Newly pulsed cells at the outer border spatially expanded the wave-permissive area (Fig. 2b–d), while those inside the cluster quantitatively increased the local density of the pulsing cells. These changes, together with the increased pulse-count in other pulsing cells, collectively increased local excitability, where the spatial gradient of the pulse-count peaked at the wave-initiating point. Indeed, the population-averaged cAMP pulsing amplitude was high near the wave-initiating point with high pulse-counts and low at the periphery with low pulse-counts, respectively (Fig. 2e, and Supplementary Fig. 3a–d). Thus, we concluded that the cluster of cells with high pulse-counts (hereafter called a pulsing cluster) in the background of homogeneously distributed non-pulsing cells underlies the locally collapsing wave in early development.

Fig. 2
figure 2

Mesoscale heterogeneity detected by the pulse-count analysis. (a–d) Spatiotemporal changes in the cAMP pulse-counts in an area (440 × 440 μm2) containing approximately 200 cells. Mesh size for analysis regions of interest (ROIs), 110 μm. Raw fluorescence images of mRFPmars showing the cell position (a) and schematics ((b–d) left) illustrating the temporal cAMP sequence at the 4th (b), 5th (c), and 10th waves (d) at cell resolution. The number of cAMP pulses experienced at the end of the indicated cAMP wave is color-coded in each cell (pulse-count). For better visibility, cells with a higher pulse-count are shown as a larger dot with a warmer color, whereas non-pulsing cells are shown as gray dots. The gray lines outline the high-pulse cluster in which the collapsing waves propagated. A spatially coarse-grained pulse-count at the ROI level is shown on the right side (b–d). (e) Population-averaged cAMP pulses in green and orange boxes in (a) ROIs with high and low pulse-counts, respectively. (f) Spatial distribution of the pulse-counts at 6:00 in half field of view (FOV). (g) Shapes of the high-pulsing clusters in (f) whose size was 0.39 ± 0.30 mm2 (Mean ± SD). (h) The wave propagation area from 6:00–6:30 roughly corresponds to 1.5 cycles of wave propagation. (i) Superimposition of the high-pulsing cluster at 6:00 (g) and wave propagation area after 6:00 (h). Black: double-negative; light orange: double-positive; purple and dark orange: single-positive for high-pulsing clusters and wave propagation, respectively. Mesh size for analysis ROIs (pixels), 110 μm.

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To further examine whether the pulsing clusters developed in other parts of the system, we performed a similar analysis on a larger scale in a spatially coarse-grained manner (Supplementary Movie 1). The length scale of collapsing waves was measured to be below 2 mm (0.24 ± 0.15 mm for the third wave and 0.52 ± 0.23 mm for the sixth wave, Supplementary Fig. 3e). We thus performed a large-scale pulse—count analysis at the cm scale in half of full-FOV (10.1 × 7.3 mm²) with a resolution of 110 × 110 μm2, where each ROI containing approximately 10 cells (Fig. 2f). At 6:00, we observed a spatially patchy distribution of localities with high pulse-counts. There were 55 pulsing clusters of different shapes (Fig. 2f–g, and Supplementary Fig. 4a), sizes (Supplementary Fig. 4b), and pulse-counts (Fig. 2f, and Supplementary Fig. 4c). The size of the high-pulsing cluster ranged from 0.048 to 1.4 mm2, containing a few tens to thousands of cells (Fig. 2g). Thus, we refer to this large and patchy distribution of high-pulsing clusters as mesoscale heterogeneity in excitability.

We examined whether the newly generated cAMP waves preferentially propagated on the established mesoscale heterogeneity as structural scaffolds. We determined the cAMP wave propagation area just after 6:00 (Fig. 2h) and compared its spatial pattern with the mesoscale heterogeneity at 6:00. The spatial patterns of the wave propagation area were similar to those of the high-pulsing clusters (Fig. 2g–h). Superimposing these two images revealed that the wave area on the high-pulsing cluster (Fig. 2i light orange) was surrounded by a wave area outside the cluster (Fig. 2i dark orange), which is in good agreement with the development of the single high-pulsing cluster (Fig. 2b–d). The fraction of existing cluster area overlapped with the newly propagating wave was > 80%, indicating that newly generated cAMP waves preferentially propagated on mesoscale heterogeneity (Fig. 2i). A significant overlap (~ 70%) between pulsing clusters and newly propagating waves was observed at an earlier time point (5:30), confirming that cAMP waves preferentially propagated on mesoscale heterogeneity throughout early development (Supplementary Fig. 5a–d). In addition, the properties of mesoscale heterogeneity might control wave initiation. To explore this, we analyzed the relationship between wave-initiating frequency and pulse-counts at the center of 10 high-pulsing clusters, identifying a positive correlation (Supplementary Fig. 5e). Specifically, clusters with fewer pulse-counts exhibited longer inter-pulse intervals (typically > 20 min for < 5 pulse-counts), while clusters with higher pulse-counts demonstrated faster pacing intervals, approaching 10 min as pulse-counts increased. These findings indicate that pulse-counts directly influences the frequency competition22 between concentric waves within their respective clusters.

These results collectively demonstrate that the pulse-count is a useful indicator for visualizing local excitability, where a high density of pulsing cells with high pulse-counts shows higher excitability than low pulsing cells. The high-pulsing cluster of a few tens to thousands of cells illustrates the mesoscale heterogeneity in excitability, underlying the spatially heterogeneous wave propagation during early development (Figs. 1 and 2i).

Wave rotation on the mesoscale heterogeneity

A close inspection of Fig. 2i shows that the propagation area of the newly generated cAMP wave after 6:00 was larger than that of the high-pulsing clusters established before 6:00. We hypothesized that the interactions among clusters might increase the wave propagation area during development. To check this possibility, we performed a detailed analysis of the wave propagation and pulse-count in a representative area containing multiple pulsing clusters (Fig. 3). At one time of middle development, when one of the pulsing clusters showed eighth round of wave propagation (Cluster-1), four distinct pulsing clusters (Clusters-1–4) demarcated by cells with zero pulse-counts were identified (Fig. 3a and g). They differed in size, shape, and pulse-count. By the 12th and 13th wave propagation rounds in Cluster-1, the spatial pattern of high-pulsing clusters interconnected with each other because newly pulsed cells filled the spatial gaps between clusters that existed at the end of the eighth wave (Fig. 3b and c).

Fig. 3
figure 3

Change of the wave dynamics on the mesoscale heterogeneity. (a–c) Pulse-count for 4,233 cells in a 2.2 × 2.2 mm2 area at the time when cluster-1 experienced the 8th (a), 12th (b), and 13th wave propagation (c), while Clusters-2–4 experienced fewer rounds of wave propagation. Cells with a higher pulse-count are shown in a warmer color, whereas non-pulsing cells are shown as gray dots. For better visibility, cells with pulse-counts of 1 and 0 are shown as small dots. The gray lines in (a) outline the cluster of highly pulsed cells at 8th wave propagation. To show the spatial expansion of clusters, the same outlines as in (a) are shown in (b) and (c), so that the newly pulsing cells outside the cluster outlines in (b) and (c) indicate cluster development. (d–f) cAMP waves at the 8th (d), 12th (e), and 13th wave propagation (f) represented by interaction links (arrows) between cells with pulse-counts larger than 0 (gray dots). The arrow indicates the direction of cAMP relaying. The temporal flow of cAMP pulsing from cluster-1 (see (g)) is shown with color-coded arrows, while the others are shown with gray arrows. (g–i) Schematic representations of the cAMP wave at the 8th (g), 12th (h), and 13th propagations (i). The wave dynamics changed from intracluster (g) to intercluster propagation (h,i). For simplicity, the morphologies of the clusters at eighth wave propagation are shown in (h) and (i). The asterisk in (i) indicates the wave-impermissive pocket. (+) indicates the wave initiation position. Green and magenta represent the wavefront and wave propagation direction, respectively. Mesh size for analysis regions of interest (ROIs), 110 μm.

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Next, we analyzed how wave propagation was affected by changes in the spatial pattern of high-pulsing clusters (Fig. 3d–f). In the eighth round of wave propagation in Cluster-1 (6:00), when four high-pulsing clusters of cells with different rounds of wave propagation were detected, the wave derived from the cell with the highest pulse-count propagated within Cluster-1. This wave collapsed at its edge because the cells outside Cluster-1 were wave-impermissive (Fig. 3a and d, and 3g). In the 12th round, the wave that originated in Cluster-1 invaded Cluster-2 due to the interconnection between Cluster-1 and Cluster-2 (Fig. 3b and e, and 3h). Interestingly, in the next round, the cAMP wave originating from Cluster-1 propagated through Clusters-2 and − 3 and reentered Cluster-1, displaying a counterclockwise rotation for few rounds (Fig. 3c, f and i, and Supplementary Movie 2). This is the moment of transition from the collapsing to a stably rotating wave. The stability of the rotating wave comes from the self-sustaining nature of such a reentry wave, where its rotated wavefront induces the next rounds of waves, making these dynamics independent of the repeated wave-initiation activities required for the collapsing waves (Fig. 3i). Notably, a wave-impermissive pocket was present at the center of the three clusters, whose pulse-counts were lower than those in the surrounding areas (Fig. 3g–i). The geometry of the three clusters forming a wave-impermissive pocket is possibly essential for generating a rotating wave.

Considering the faster wave initiation pacing for higher pulse-counts (Supplementary Fig. 5e), the establishment of invading and rotating wave also is likely influenced by mesoscale heterogeneity. Before the invading wave was initiated in cluster-1 (i.e., 12th), it exhibited faster pacing at 10.5 and 10 min intervals for its 10th and 11th waves, compared to cluster-2 with 8 pulse-counts, which showed intervals of 21 and 17 min for its 7th and 8th waves, respectively. Thus, the initiation of invading wave in cluster-1, rather than in cluster-2, represents an atypical example of frequency competition between the center of two concentric waves (cluster-1 vs. cluster-2). Once the rotating wave was established across clusters-1, -2, and − 3, wave initiation at the original centers of these clusters was suppressed, likely because the higher frequency of the rotating wave overrode the lower frequencies at these centers23.

In summary, these observations and analyses suggest that changes in the structures and properties of mesoscale heterogeneity, particularly the interconnection of the high-pulsing clusters having different pulse-counts generated a wave circuit with a low-excitability core, resulting in sustained wave rotation. These are the first direct observations suggesting the functional importance of mesoscale heterogeneity in controlling wave pattern formation in our model system. Interestingly, the observed wave rotation and reentry behave similarly to the dynamics of functional reentry in cardiac physiology2, which is the source of spiral waves. In the following section, we ask possible involvement of functional reentry in spiral formation in developing D. discoideum cells.

Spiral formation by functional reentry

We next investigated how dominantly functional reentry is involved in spiral formation (Fig. 4). Past theoretical studies of excitable systems have suggested that spiral waves can be generated by several independent mechanisms, such as wave–pulse interaction (vulnerability)13, wave–obstacle interaction (vortex shedding)24, and wave rotation around the low-excitability core (functional reentry)7. The emergence of wave fragments and subsequent rotation occur differently in these processes and are specifically associated with ectopic pulses for vulnerability, a non-excitable obstacle for vortex shedding, or a low-excitability core for functional reentry. We retrospectively and prospectively analyzed cAMP wave emergence and rotation to differentiate the involvement of these mechanisms in our model system. The analysis was applied to half FOV, which was sufficiently large to detect the development of multiple spirals. The number and trajectory of rotating waves were analyzed by detecting their tips, as is commonly performed in the analysis of spiral waves2.

Fig. 4
figure 4

Spatial and temporal distribution of functional reentry. (a,b) Schematics of reentering and non-reentering wave rotations around the low-pulsing pocket. (c) Quantitative analysis of rotational wave dynamics in 7.3 × 10.1 mm2. The location of reentering (magenta) and non-reentering (cyan) rotations during 6:15–9:00 were imposed on the spatially coarse-grained representation of pulse-count at 7:30. Scale bar, 0.5 mm. (d) Numbers of observed rotational dynamics during 6:15–9:00.

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At 10:00, which corresponds to late development, we identified nine rotating spiral wave tips that persisted stably for the next few hours. Retrospective tracking of the spiral wave tips revealed that all spiral waves developed from the wave fragments were closely associated with the low-pulsing pockets generated by 8:30 (Supplementary Fig. 3 and Supplementary Movie 1). Spiral development from wave fragments in association with ectopic pulsing or unexcitable obstacles was not observed, indicating that functional reentry but not vulnerability or vortex shedding is the primary mechanism for spiral formation in this system.

To further understand how rotational waves culminate in spiral waves, we performed a prospective analysis by classifying the rotation dynamics into reentering and non-reentering based on their rotation angle (Fig. 4a and b). In half FOV during 6:15–9:00, we identified 174 wave rotation events, half showing a clockwise rotation and others showing a counterclockwise rotation. The majority were non-reentering rotations (θ) (180° < θ < 360°, 136 cases) rather than reentering ones (θ > 360°, 38 cases). To determine whether these rotational waves were associated with low-excitability cores in mesoscale heterogeneity, we depicted the identified wave rotations on a coarse-grained pulse-count map representing the spatial organization of mesoscale heterogeneity in excitability (Fig. 4c). The circular trajectories of the wave tips were closely associated with the low-pulsing pockets formed by two to four high-pulsing clusters (Fig. 4a–c). Among the 38 reentering rotations, only nine culminated in spirals. The others disappeared in up to three rounds of rotation owing to the pairwise collision of the clockwise and counterclockwise rotating wave tips, a known process called annihilation. At the low-pulsing pockets, most reentering rotations were spatially overlapped with trajectories of multiple non-reentering rotations. These observations indicated that the spiral waves developed from many reentering rotations, which were also seeded by many non-reentering rotations.

To determine when functional reentry occurred during development, the number of reentering (N = 38) and non-reentering (N = 136) rotation events was plotted against time after starvation (Fig. 4d). Wave rotation was observed at 6:00 without reentry. Reentry was first detected at 6:30 and peaked between 7:00–7:30. No further attempts at reentry were observed after 8:30, suggesting the presence of a critical period for establishing functional reentry.

These observations were reproduced in four independent experiments at the same (N = 3) and double the cell densities (N = 1) (Supplementary Fig. 7). We thus conclude that spiral formation in developing D. discoideum is driven by functional reentry during middle development across the tested cell densities. An invariable spiral formation, a unique feature of this system, can be realized through multiple trials of non-reentering and reentering rotations.

Mesoscale heterogeneity in excitability is essential for spiral formation

To determine whether mesoscale heterogeneity in excitability is essential for functional reentry, we conducted two types of perturbation experiments: classically established phase resetting25,26 and the newly introduced spatial resetting (Fig. 5 and Supplementary Movie 3).

Fig. 5
figure 5

Loss of spiral formation on the spatially homogenized excitability. (a,b) Wave patterns before and after resetting. Phase and spatial resetting were performed at the critical period for spiral formation (a) and post-critical period (b), respectively. Composite image of two successive pulse-enhanced channels with 1 min delay. Images immediately before resetting and at the third recovery from resetting. Scale bars, 2 mm. (c) Average number of wave tips for reentering (filled) and non-reentering (open) wave rotations at the third recovery cycle after resetting. N = 2 experiments (8, 5 and 18, 20; for reentering and non-reentering wave rotations in phase resetting at a critical period, respectively. 0, 2 and 4, 4; for reentering and non-reentering wave rotations in spatial resetting at a critical period, respectively). Error bars, standard deviation. ND, not detected.

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First, phase resetting was performed by synchronous cAMP stimulation to transiently erase the existing waves while leaving the spatial distribution of excitability unaffected. By examining the emerging wave patterns, we determined whether a driving force for the wave rotation is present at the time of phase resetting. As demonstrated previously25,26, phase resetting in late development (9:00) completely erased the existing waves and led to the reappearance of concentric, but not rotational, waves (Fig. 5b top). Phase resetting in the middle development (6:15), when functional reentry was started (Fig. 4d), resulted in the reappearance of both reentering (6.5 tips/full-FOV, N = 2) and non-reentering (19 tips/full-FOV, N = 2) wave rotations (Fig. 5a top and 5c). These results indicate that the driving force for functional reentry is temporally specific to middle development and unaffected by phase resetting.

Functional reentry occurring on mesoscale heterogeneity in excitability was temporally specific to middle development (Fig. 4). This suggests that mesoscale heterogeneity during middle development may be essential for functional reentry. To check this possibility, we performed spatial resetting by detaching the cells from the culture dish through intensive pipetting, where the cells resumed cAMP pulsing after settling in random positions on the dish. While the effects of spatial resetting are parallel to those of phase resetting due to buffer mixing, we can still ask whether the rotational wave appears on a salt-and-pepper pattern of excitability at the cell resolution rather than on a large and patchy excitability at the mesoscale by comparing the results of both spatial and phase resetting. When spatial resetting was performed at the middle development (6:15), the reappearance of both reentering and non-reentering wave rotations was significantly reduced to 1/5 of that observed with phase resetting (Fig. 5a bottom and 5c). These perturbation experiments directly confirmed that the mesoscale heterogeneity in excitability during middle development is essential for wave rotation in functional reentry.

Discussion

Using trans-scale imaging and pulse-count analysis of intercellular communication, we directly demonstrated the presence and functional importance of mesoscale heterogeneity, which transiently appears during the pattern formation of a living excitable system. Our study highlights the unexplored aspects of mesoscale heterogeneity beyond the well-studied microscale and macroscale heterogeneities.

This study represents one of the first direct visualizations of the spontaneous spiral wave formation, a phenomenon whose mechanism has long been a mystery. We found that collapsing waves transit to rotating waves on mesoscale heterogeneity in excitability, where low-excitability pockets were self-organized in developing D. discoideum cells. This process, classically known as functional reentry27, has been implicated in spiral formation in cardiac tissues. A previous study using a monolayer sheet of cardiac myocytes demonstrated that the reentry-related waves are more easily induced in cell sheets with lower densities, where spatially heterogeneous electrical conductivity was developed28. The similarity between this medically relevant setup and our system suggests that mesoscale heterogeneity-driven functional reentry may be a universal mechanism for spontaneous spiral wave formation in living excitable systems. Owing to the clinical significance of spiral waves, which are involved in various diseases, such as life-threatening cardiac fibrillation2,29, neuronal epilepsy1, and progressive dermal inflammation30,31, comparative studies on the mechanisms of spontaneous spiral formation in other systems could inform effective prevention and care strategies for disease-associated spirals.

Moreover, the mechanism underlying the development of mesoscale heterogeneity must be addressed. Microscale heterogeneity in cellular excitability, which is intrinsic to the system during early development, may play a crucial role. As observed in pulsing cluster development, the properties of cellular excitability for wave initiation and signal relay likely influence mesoscale heterogeneity, thereby affecting the number, shape, and gradient of excitability within each cluster. However, determining cellular excitability before cAMP pulsing, which develops through a pulse-dependent positive feedback mechanism, is technically challenging owing to the lack of suitable molecular markers. Therefore, mathematical modeling could be invaluable for enhancing our understanding of the self-organization mechanism of mesoscale heterogeneity in future studies.

Beyond molecular specificity, the significance of mesoscale heterogeneity might be universal in developmental pattern formation across diverse multicellular systems. Spatially patchy distributions of cells with distinct properties have been observed in other systems, such as biofilms32, embryogenesis33, and tumorigenesis34, and their roles in the overall system dynamics are just beginning to be recognized. To deepen our understanding of mesoscale heterogeneity, cellular properties at the microscale and system dynamics at the macroscale35,36 should be analyzed simultaneously within the context of trans-scale studies15,36, as system-level properties can emerge from functional interactions among cell clusters at the mesoscale rather than the cellular scale. Advancements in imaging strategies, omics analysis, and data processing can accelerate these investigations.

Methods

Molecular biology

cDNA encoding the mRFPmars–Flamindo2 fusion protein (Red-FL2), for which codon usage was optimized for D. discoideum, was constructed using the InFusion cloning system (TAKARA) and cloned into pDM304 and pDM35837. The resulting plasmids, pDM304_Red-FL2 and pDM358_Red-FL2, were deposited at the Dicty Stock Center.

Cell culture

The axenic strain Ax2 was cultured and transformed as described previously38,39. For transformation, the 1 × 107 cells were washed and suspended in each mL ice-cold EP buffer (6.6 mM KH2PO4, 2.8 mM Na2HPO4, and 50 mM sucrose, pH 6.4). An 800-µL cell suspension mixed with 10 µg pDM304_Red-FL2 in a 4-mm-wide cuvette was electroporated (two 5-s separated pulses with 1.0 kV and a 1.0 ms time constant) using a MicroPulser (Bio-Rad). These cells were plated on 4–6 90-mm plastic dishes with HL5 medium, incubated at 22 °C for 18 h under non-selective conditions, and cultured with 10 mg L− 1 G418 (Wako). After 4–7 days, high Red-FL2-expressing colonies were manually selected. Some colonies were subsequently transformed with pDM358_Red-FL2 and cultured with 35 mg L−1 hygromycin (Wako) and 15 mg L− 1 G418. Clones with high Red-FL2 expression and low heterogeneity were screened.

Imaging

Cells expressing Red-FL2 were maintained in HL5 medium on a 90-mm plastic dish at < 1 × 106 cells per dish. The development was initiated by three washes with development buffer (5 mM Na2HPO4, 5 mM KH2PO4, 1 mM CaCl2, and 2 mM MgCl2, pH 6.4). Then, approximately 850 cells per mm2, the highest density allowing reliable single-cell tracking, were then plated on a 35-mm glass-bottom dish. The live cells were imaged using a custom-built imaging system equipped with a single CMOS image sensor and LED illumination15. The cell density in the reset experiments was doubled to improve visibility. At least two independent experiments were performed to confirm reproducibility.

Image analysis

Ratio images of background-subtracted and spatially smoothed channels for mRFPmars and Flamindo2 were enhanced for pulsing cells assisted by supervised machine learning using AIVIA software (DRVISION Tech.). The image field was subdivided into 12,236 analysis ROIs (133 × 92 matrices of 110 × 110 μm2) and peaks on time series data for every ROI were detected using Mathematica. The wavefront and cumulative pulse-counts were analyzed in each analysis-ROI containing approximately 10 cells. The image reconstruction from these data was performed with a custom-built analysis pipeline using Excel, Mathematica, Matlab, and Fiji software. For peak detection, the sensitivity was tuned to detect > 1 pulsing cells/analysis-ROI. After obtaining a peak table for 900 frames of 12,236 analysis ROIs, those from the left half of the full-FOV were manually corrected to detect one pulsing cell/analysis-ROI with a ΔRatio of Red/FL2 (ΔR) > 5%. The pulse-count analysis at cell resolution was performed by manually tracking 4,233 cells in a 2.2 × 2.2 mm2 area for 500 frames corresponding to 4:00–8:10. High-pulsing clusters (Fig. 2g) were manually extracted and segmented after considering wave-initiating points. To quantitatively compare the cAMP pulse magnitude among cells (Supplementary Fig. 2a), we performed two-step normalization for ΔR. Considering the 1:1 stoichiometry of red and yellow Red-FL2 signals among cells, we pre-normalized the baseline ratio to 1.0 to cancel the different baseline ratios of distantly positioned cells caused by a slight imbalance of illumination strength over the image field. The maximum pre-normalized ΔR for > 300 cells by prolonged stimulation with 20 mM 8-Br-cAMP, a cell-permeable cAMP analog, was 5.540. Finally, we normalized pulse data from all examined cells by transforming the minimum (1.0) and maximum (5.5) pre-normalized ΔR to 0 and 1, respectively.

Phase and spatial resetting

Phase resetting of the cell population was performed by bath application of cAMP (final concentrations of 20 and 50 nM for early and late development, respectively)25,26. Spatial resetting was performed by intensive buffer mixing via pipetting, which detached the cells from the culture dish. These resettings were done at 6:15 or 9:00, corresponding to the pre- and post-critical periods of spiral formation, respectively. Both resettings cause forced excitation of the entire cell population, causing the cells to enter the refractory period. After recovery from the refractory period (15–20 min), the cells restarted cAMP pulsing to generate cAMP waves. The entire wave dynamics, including before and after resetting, was imaged every 30 s.